While my pre-calc class has been digging hard into the world of technology and inquiry for their learning, I am feeling kind of guilty and kind of not-guilty for doing direct instruction and manual calculation in pre-algebra.
We’re working on a unit on decimal operations. Decimal operations are included in the 5th grade Common Core standards, but my assessments in our previous unit on positive/negative numbers told me most students aren’t 100% masterful with them yet.
I have only been a *tiny* bit constructivist with this unit. We did a brief activity using base-10 blocks to understand place value. I gave the students a pre-assessment on adding and subtracting with decimals, and based on the pre-assessment I could identify maybe 15 students over all three sections that needed some help with those operations. I gave most of the class some time on Khan Academy while I did small-group instruction with those kids. We just worked problems and focused on proper carrying/borrowing/place value. Every single kid was familiar with the basic idea, they just had some gaps when it came to the algorithm. The small-group work was just what they needed.
For decimal multiplication, I gave the students a worksheet I made up with some problems that had patterns in them.
I did not teach the students how to multiply numbers with decimals first… I just wanted to see if they could use the patterns in the worksheet to infer how big the numbers would be. We went over the answers together and then I asked the class if there’s a rule they started to pick up that told them where to put the decimal in the answer. Every class had multiple students that said something to the effect of: “however many decimals are in the problem, that’s how many are in the answer”. All we had to do was formalize that into a rule and then practice it.
As a unit project, I taught the students about how to write an invoice if they own a small business and need to charge their customers for items they buy. Without doing anything constructivist yet, I just taught the kids how to convert a percent into a decimal and then multiply to calculate discounts and tax. Creating an invoice involves decimal addition, subtraction, and multiplication, so it’s a great assessment to the unit.
For the students’ unit assessment, they will create their own virtual store that has multiple items I could buy, some made-up coupons involving percent discounts, and a made-up tax rate somewhere between 1% and 12%. I’ll shop at each student’s store and pick out some items and coupons, and they will have to make me an invoice by hand.
Now for the reflection on this unit so far: I have really mixed feelings about the unit’s topic, manual calculation of decimal operations. I am totally on board with the understanding that in a computer-driven world like ours, they can get by without knowing it. They can actually thrive without knowing how to do these calculations manually.
Why, then, does this feel so satisfying? Kids thanked me for teaching it to them. An intern volunteering in my classroom commented “this is really cool. I never actually learned how to do this. It’s great.” I’m reminded that I *can* plug a sudoku puzzle into a computer program and have it solved for me, but it’s satisfying to work through the sudoku puzzle myself and know I’m just as smart as a computer. Plugging numbers into a calculator is just as much of a rote task as large-number multiplication, but doing the math by hand, in the right mindset, gives you ownership of the puzzle.
Is that why manual math has value? Does it have value? I do not plan on emphasizing this kind of manual calculation once this unit ends – when we get into rates, ratios, equations, and expressions, we’ll use computerized tools as problem-solving aids most of the time to free up our working memory.
The other part of these lessons I’ve wrestled with is the direct instruction vs. constructivist. The lessons started out with a little pattern-finding and then I just taught the kids how to do the math. Over time, I have toned down the inquiry in some situations and I now believe it’s possible to overdo inquiry. Sometimes you are done trying to ferret out the method for yourself and you just want someone to show you how to do it. Some struggle is healthy but not unlimited struggle.
I would not have taught like this six years ago. If there’s a scale where 0 is pure direct instruction and 10 is pure constructivism, I would say this set of lessons is like a 2 or 3. What should be your criteria for determining when you do inquiry and when you do direct instruction? Lots of questions I don’t have easy answers to.
But this set of lessons, this unit, felt satisfying to teach and I expect good results from our little shopping assessments. I don’t have any regrets.
Just blogging to report on my two big initiatives for the week: finishing up a mini-project on trigonometry using the micro:bit accelerometers, and a week on anti-racism in my sixth-grade advisory.
The mini-project finished up SO well. We started last week by assembling micro:bit inclinometers and programming them to report out the angle measured by using trig ratios. This week, we took measurements with them and did the math. The students measured the height of a neighboring building by standing exactly 800 centimeters away and sighting the roofline with their micro:bit devices. It’s been a cold week so we rushed inside to do the math, and the kids calculated a building height of around 39 feet. We probably got in the ballpark.
One commenter on my blog suggested we try another method, sighting an object from a distance, then backing up a known distance and taking another reading. This ended up being a cool application of a system of equations. Great suggestion!
Even though learning to solve systems of equations feels like drudgery at first, it’s one of the more useful algebra concepts I have learned. In my decade as an engineer, I modeled many situations involving systems.
The kids did a great job and seemed to enjoy the challenge of the activity.
The other day, I blogged about the very strange situation involving the Catholic school teens in DC mocking other protesters near the Lincoln Memorial. With the MLK holiday coinciding with that weekend, I really felt the need to do a lesson series on racism in my advisory class. They are sixth-graders, and I decided to survey them first – I’m glad I did. Only one of the kids had even heard of the incident with the protestors, which I found fascinating. What we perceive as going viral doesn’t reach all age groups equally. I decided to leave the incident off the lesson plan and focus on basic, age-appropriate lessons on racism.
We started with a few circle prompts about times they have ever seen someone treated unfairly because of their race, skin color, or religion. I asked them about their perceptions of bullying at our small school. Almost all the students feel that at our school, bullying is not really so much the issue. Some students can be rude or insensitive, but they understood the difference between rudeness and bullying.
I talked about how it’s healthy that in our lifetime, it’s not socially acceptable to be outwardly racist. People in our community tend to call each other out on it. Racism tends to take the shape of the many small ways in which your life is made a little easier or a little harder because of your race. We read through the list created by Peggy McIntosh in the Invisible Knapsack. Some of the students were able to chime in with ways in which privilege tends to show up at school – for example, if a girl hears “you’re pretty smart! Girls can be smart can’t they?” or if you get in trouble with the principal and a little voice makes you wonder if it’s because of the way you look.
Today, we started watching “A Class Divided“, the story of Jane Elliott’s brown eyes / blue eyes discrimination experiment in her third-grade classroom. The kids find it fascinating. It’s a very good and comprehensive introduction to what discrimination is and what it does to people. As a sixth-grade lesson, I think it’s a solid foundation to build on. I’m interested to debrief with the kids tomorrow to get their thoughts on it. By the end of class today, they were begging me to try the experiment at my school – to bring collars and let them try it. I think they’re intrigued at the idea of lording it over their classmates, but also wondering if the collars would make them the same nasty people that the third-graders became when they were empowered over their lower-class friends.
Jane Elliott is a nationally-known advocate for racism education now, and I told the students I have seen her do this experiment on adults and it’s still just as powerful. She can make adults cry. (I won’t show that video to the kids because there is swearing… but man!)
At some point, one of my students raised his hand and said “Donald Trump is racist.” I sat in it for a moment and then said “Yes. He is.” Other kids wanted to chime in. I allowed a few comments. I ended with “He says racist things. It’s not OK.” And I moved on. I have heard arguments that Trump says the things he says to provoke crowds, or build his “message”. If you say racist things to provoke or build your message… you are a racist. I would never allow one of my students to say the things the president says. That behavior needs to be named.
It’s not a ton of lessons, but the students have been engaged and receptive, and I think it got us off to a good start when it comes to understanding these themes.
I have a few things to say about racism. I’m grateful, in an odd way, for the viral video of the boys wearing MAGA hats and mocking the indigenous drummer in Washington DC. You know the one.
It hit me hard because I recognize those kids. Not these exact teenage boys. But the smiles, the color, the jumping, the shouting, and the hats. They look just like my kiddos in northern Colorado. I worked with over 1000 kids just like them last year at my suburban public school.
I am amazed that we live in an era in which the president’s name and signature slogan are used as racial taunts. And yet here we are. Over the past couple of years, it’s been shocking to see kids act openly racist to their classmates simply by yelling TRUMP in the face of a Latino classmate with whom they already don’t get along. Or by screaming TRUMP’S GONNA BUILD THAT WALL in a crowded hallway and then ducking into a bathroom. Amazing that a kid can wear a MAGA hat to school and just revel in the high-fives from half of the students and the glares from the other half, and just bubble over with joy at the chaos they’ve wrought around them. Because of a President’s slogan.
I have NOT handled it well for the past couple of years. I have tons of excuses. I really wasn’t prepared for students to act racist to one another. I’m not trained in strategies for ending it. If I intervene, how much of my motivation is based on ending bullying and harassment, and how much of my motivation is based on not liking the politics of the kid’s parents? What if the parents call me and accuse me of being anti-Trump? Am I prepared for that call? If not, maybe I should scale down my response until I can really understand what’s driving me. Is the kid really being racist or am I overreacting? Maybe I’ll console the kid who’s being teased and avoid confrontation with the bully. I swear I had these thoughts. I’m not proud of them. I do want to respect everyone’s political differences and suburban parents can be intimidating. But come on – I was looking the other way when there was obvious racial bullying going on, disguising itself as politics.
The incident with the Native American drummer was fascinating because it happened on a school outing. I found myself empathizing with the school and internally making excuses for them. I have been there. Teenage boys do stupid things. My own students have these same impulses. And while we’re passing the buck, I believe the president’s own racism is at the heart of these issues, absolutely. The man has claimed Central American asylum seekers are diseased. He tried to make his case for a wall based on the presence of possible prayer rugs in the desert. He made fun of Wounded Knee. These are only from the last few weeks – the man is not politically incorrect, he is outright racist, period.
I stopped empathizing with the school when I realized I needed to stop making excuses for myself. It’s irresponsible of me to not even try to clean up the mess when his words set fire to my student community. I can’t let these moments go by. Kids need to know their words and actions are not ok, and they need to be called out for what they are. No more “you guys need to just stop and walk away”. It needs to be “I heard you say BUILD THE WALL as a way of harassing another student. Racial intimidation is serious and we need to take some time out to deal with this.” Covington Catholic School – and me, and my former school, and my current school, need to have a moment of reckoning and realize we have to deal with these forces that have been thrust upon us. We have a racist president who inspires kids to be racist. This is the world we live in. That nonsense HAS TO STOP and we need to not back down because we’re worried about offending their parents.
And if any of you reading this have suggested tools, techniques, articles or whatever to help someone learn the best way to correct racism and right the ship, please send them my way. I believe in restorative discipline and want to not just end racist, harassing, and bullying behavior but create a positive, empathetic community.
This year I am teaching pre-calculus for the first time, and I am committed to doing projects with my students as much as possible. Last semester we created a parabolic trough solar oven and made holiday cookies for students. This semester I decided to start with a unit on trigonometry, and I happened upon an interesting project via Twitter that showed someone sighting a distant object and using a micro:bit’s accelerometer to calculate tilt and thus how tall the object was. What a cool application of computing and trig. I decided to try and create the project for my class.
I decided to spend some time actually creating the thing. I started by attaching the micro:bit to a cardstock tube. The tube could be used to sight the top of a tree or building. We would try to keep the micro:bit on the same side and simply adjust the tilt until an object was sighted through the tube.
I played with different programming languages and decided to use Python, because it had a robust library of math functions. I started with a simple program to just fetch the accelerometer readings when you push a button.
I found that if the tube is held level, the “x” reading was close to 0, the “y” reading was close to a maximum of 1024, and the “z” was close to 0. If I held the tube pointing straight up (90 degrees), “x” was -1024 and “y” was close to 0. “z” remained close to 0. So as you tilt the micro:bit, the “x” accelerometer goes from 0 to -1024 while the “y” accelerometer mirrors it and goes from 1024 to 0.
I did a little searching to figure out how to convert accelerometer readings into an angle of inclination. There are a lot of different formulas out there – probably all correct. One source I found had a very simple equation:
So basically the angle of inclination is the inverse sine of the ratio of the “x” accelerometer to 1g. I had a hard time visualizing why the ratio x/1g would be equivalent to an opposite / hypotenuse, but it started to make sense when I realized the forces at work are really similar to the kinds of forces on an object that slides down an inclined plane.
In the diagram, the parallel force is analogous to the reading on the “x” accelerometer. The perpendicular force or “normal” force is analogous to the reading on the “y” accelerometer. Fgrav is basically 1024, the reading you get when there is a full 1g on an accelerometer.
This triangle is similar to the triangle made by the inclined plane. I made a little sketch that maybe shows this more clearly?
So basically the formula above, the simple inverse-sine operation, works because your angle of inclination is congruent to the angle opposite the “x” acceleration vector. You can find that angle by finding inverse-sine of “x” to “force of gravity”, 1024.
I wrote another Python program that did this math and reported out the angle, and it seems to be reasonably accurate.
Once you know the angle, if you know how far away you are from your object, its height can be found this way.
tan(theta) = height / distance
distance * tan(theta) = height
Easy peasy! This assumes you’re sighting from the ground. We may find we have to adjust for eye height. We can do that. Time to create the student-facing activity.
I put together this packet for the students. My class is super tiny so the kids can go through it as one group. For a larger class I would make groups and do lots of catch and release.
Here’s how it went.
We watched the video on the biltmore stick. Students gave me hypotheses around why it worked, and we talked about potential sources of error.
I told the students that with modern technology, we should be able to make a decent height-finding tool. I introduced the micro:bits to them and told them about some of the features. They’re the first kids in my school to use the micro:bits, and they were ENCHANTED by them. You turn them on and they show messages and images, they play a game, and then they tell you to get coding. How fun! The students had a zillion questions about what else the micro:bits can do and how they worked. After the excitement faded just a little, we talked about accelerometers and how they worked, and the students started working through the packet.
I hoped they would be able to struggle through most of it up until they had to write their procedure in Python, but of course that isn’t how it went. We ran up against several big conceptual roadblocks.
- The idea of the x, y, and z-axis accelerometers BLEW THEIR MINDS. It was really tough to visualize which axis was which, and the students twisted and turned the micro:bits every which way. They had a very tough time being systematic about turning the micro:bit on just one axis to narrow down which accelerometers were changing. I hoped they would be able to sort out which axis was which on their own. They could not, and they got frustrated really quickly. I broke down. I just told them which axis was which and what the max and min values were. I have to admit this has been a struggle for me as well. Visualizing the three accelerometers is a challenge and I probably would have felt the same way in this activity.
- The accelerometers are really sensitive. One moment you set the micro:bit level and get a reading of 0. Another moment the micro:bit seems to be in the same position but your reading is -92. Another moment it’s 16. The text scrolls slowly so you don’t really appreciate what those readings look like in the moment. It was hard, then, to ferret out what the max and min values were. They floated around.
- I really thought with their geometry background the students would visualize the similar triangles really quickly. They did not. Looking back, I remember feeling frustrated and like my mind was a little blown when I learned about forces on an inclined plane. So I should have been ready for this. But the whole idea that a force of 1g directed toward the ground could be broken up into the x and y components on the accelerometers, and that they didn’t add up but rather made legs of a right triangle… WHOA. There was yelling. There was almost crying. Emotions were high. Eventually they did seem to understand but I am going to have to do some good formative assessment next class to see what they actually got.
I have a TINY class, only 5 students, and so the yelling and the emotion was totally manageable, but I am SO glad I did not go through this with a bigger class. I would have done a lot more pre-work on gravity, inclined plane forces, and similar triangles.
Today, after all of our drama, the students wrote programs to calculate the angle of inclination and strapped the micro:bits to paper tubes. Next class, we’ll go outside and take measurements. One of my students found an alternate method of measuring the angle that was something similar to this.
It seems to use a distance-formula calculation instead of the force of gravity and it’s interesting how it uses all three axes. I’ll let her try it and see if her results are similar.
I’ll take some pictures of the results of the experiment and hope I get to do this again with a future group! I feel like with better pedagogy this would be a really great activity!
Well, I have successfully (?) survived my first semester at a startup school. It wasn’t always pretty, I’m a little bruised, but I have some changes I am ready to make. Here they are.
- I am determined to get better quality work from my students. I think the students I had in my venture project class enjoyed the course and made creative projects, but I was disappointed in the quality if I’m being very honest. Their communication products were especially poor. Without a good poster, presentation, or research paper to accompany your project, it kind of looks like a 4th grade diorama. And some of the projects even by the 8th – 10th graders kind of looked like 4th grade dioramas. As a staff, we came to some common understandings about what makes good research – and I will also spend some time digging up exemplars and industry examples that the kids can look to for inspiration. Communication will be a full third of their “grade” for their projects and I need to leave enough time at the end of the semester to work on this piece instead of squealing into the exhibition still throwing hot glue on a project.
- I’ve carved out my space. I really needed part of a classroom to call my own, and I kind of took over the makerspace in the school. I needed to feel like I had some ownership over my little section of the building, in order to take pride in the work that takes place there. I hung posters, organized all of the shelves, set up the 3-D printers and sewing machines, carved out some space for student projects, and I made little journals for the kids. We were in such a rush to open the building that a lot of this basic stuff just got left. Living in a pile of clutter and unfinished setup is stressful on its own. Now things are organized and set up and I am calmer. I also recognize that claiming space is a big part of what gives the kids ownership in the school, so I’ll look for ways the students can display their work and label their own areas.
- I set up protocols for the makerspace. Because of our disorganization last semester, a lot of our equipment was used carelessly. Students would just grab anything from jumper wires to knitting needles to impact wrenches and either use them incorrectly or fidget with them. For any equipment that is either dangerous or delicate, I’ve set up a 3-tier badging system. A level 1 badge means you can follow a basic tutorial to use the tool with support. A level 2 badge means you can use the tool independently. A level 3 badge means you can troubleshoot the tool and also teach others to use it. When you have reached level 3, you can use the tool without staff supervision and also teach other students how to use it. I posted lists of our students that have level 3 badges next to everything – the 3D printers, the sewing machines, the Glowforge, the soldering irons, the podcasting system. Currently the lists are empty, but as students level up, they can start using the tools independently and showing their friends what to do.
- I think I have the start of a really fun venture project. This quarter’s venture project asks the question “What does it mean to be a Maker?” I toyed with the idea of having the students make inventions that were purposeful and noble, but when I reflected on how I got started as a “Maker”, I realized I only started being able to make purposeful things after I had spent some time tinkering and making a bunch of crappy things. I needed to have the permission and courage to use tools to make things that were lousy but fun and interesting to me. So the secondary title is “Useless Inventions”. We’ll learn about Arduino and micro:bits and 3D printing, and we’ll visit Fort Collins’ Creator Hub, and as the kids’ final project they will make their very own Useless Invention. It should be joyful and interesting but it’s perfectly OK if it’s crappy. Making a few crappy, entertaining things will help them build resilience and grit in a non-threatening way. The inspiration is this TED talk by Simone Giertz. When I’ve pitched this idea to students, they have been VERY excited about it and rattled off a list of a few things they’d like to make. A jellybean cannon. A fantasy carousel. A jump-scare machine. A virtual aquarium. I want them to appreciate the Maker Mindset and get involved in the open-source community. The students will share and publish their projects, giving credit to whatever inspired them. They can publish tutorials on a site like “instructables”, or even make YouTube videos. I’ve got a formal planning document but that’s the gist of the project.
It’s late and I will write more about venture project and what I expect from math this quarter… I’ll do that part later. I see my first group of students tomorrow and I ought to get some rest.